musl/src/math/log2.c

/* origin: FreeBSD /usr/src/lib/msun/src/e_log2.c */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */
/*
 * Return the base 2 logarithm of x.  See log.c and __log1p.h for most
 * comments.
 *
 * This reduces x to {k, 1+f} exactly as in e_log.c, then calls the kernel,
 * then does the combining and scaling steps
 *    log2(x) = (f - 0.5*f*f + k_log1p(f)) / ln2 + k
 * in not-quite-routine extra precision.
 */

#include "libm.h"
#include "__log1p.h"

static const double
two54   = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
ivln2hi = 1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
ivln2lo = 1.67517131648865118353e-10; /* 0x3de705fc, 0x2eefa200 */

double log2(double x)
{
	double f,hfsq,hi,lo,r,val_hi,val_lo,w,y;
	int32_t i,k,hx;
	uint32_t lx;

	EXTRACT_WORDS(hx, lx, x);

	k = 0;
	if (hx < 0x00100000) {  /* x < 2**-1022  */
		if (((hx&0x7fffffff)|lx) == 0)
			return -two54/0.0;  /* log(+-0)=-inf */
		if (hx < 0)
			return (x-x)/0.0;   /* log(-#) = NaN */
		/* subnormal number, scale up x */
		k -= 54;
		x *= two54;
		GET_HIGH_WORD(hx, x);
	}
	if (hx >= 0x7ff00000)
		return x+x;
	if (hx == 0x3ff00000 && lx == 0)
		return 0.0;  /* log(1) = +0 */
	k += (hx>>20) - 1023;
	hx &= 0x000fffff;
	i = (hx+0x95f64) & 0x100000;
	SET_HIGH_WORD(x, hx|(i^0x3ff00000));  /* normalize x or x/2 */
	k += i>>20;
	y = (double)k;
	f = x - 1.0;
	hfsq = 0.5*f*f;
	r = __log1p(f);

	/*
	 * f-hfsq must (for args near 1) be evaluated in extra precision
	 * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
	 * This is fairly efficient since f-hfsq only depends on f, so can
	 * be evaluated in parallel with R.  Not combining hfsq with R also
	 * keeps R small (though not as small as a true `lo' term would be),
	 * so that extra precision is not needed for terms involving R.
	 *
	 * Compiler bugs involving extra precision used to break Dekker's
	 * theorem for spitting f-hfsq as hi+lo, unless double_t was used
	 * or the multi-precision calculations were avoided when double_t
	 * has extra precision.  These problems are now automatically
	 * avoided as a side effect of the optimization of combining the
	 * Dekker splitting step with the clear-low-bits step.
	 *
	 * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
	 * precision to avoid a very large cancellation when x is very near
	 * these values.  Unlike the above cancellations, this problem is
	 * specific to base 2.  It is strange that adding +-1 is so much
	 * harder than adding +-ln2 or +-log10_2.
	 *
	 * This uses Dekker's theorem to normalize y+val_hi, so the
	 * compiler bugs are back in some configurations, sigh.  And I
	 * don't want to used double_t to avoid them, since that gives a
	 * pessimization and the support for avoiding the pessimization
	 * is not yet available.
	 *
	 * The multi-precision calculations for the multiplications are
	 * routine.
	 */
	hi = f - hfsq;
	SET_LOW_WORD(hi, 0);
	lo = (f - hi) - hfsq + r;
	val_hi = hi*ivln2hi;
	val_lo = (lo+hi)*ivln2lo + lo*ivln2hi;

	/* spadd(val_hi, val_lo, y), except for not using double_t: */
	w = y + val_hi;
	val_lo += (y - w) + val_hi;
	val_hi = w;

	return val_lo + val_hi;
}